When I'm cutting something for a small size, I fold enough fabric (selvedge to centre) to cut the piece and no more. Usually it's only the larger sizes that you have to fold the fabric right to the middle. And I pin it in place if it's not the type of fabric that will just sit there. If it slides at all, it gets pinned. For the second piece (like the top being cut in the diagram on the link), I'll check the pattern to see where it matches and/or where the interesting bits are. If I'm going to have a leftover, I want it where I want it. Sometimes the selvedge edge is the best because you can make a belt/scarf from it. So, the second piece gets cut either from folding over just after when I finished cutting, or like the illustration. Just check your fabric width to make sure everything fits and you get the pattern repeat where you want it. It all depends on whether there's a pattern repeat that needs matched and how big the size is. Of course, I'm a notorious fabric squeezer who just got 3 pairs of drop waist pants for my short skinny DD from 3 meters of 60" fabric.

The grain will be straight as long as your fold is straight. To insure the fold is straight you would measure from the selvage to the fold at one end of the fabric then again at the other end - as long as both measurements are the same your grain should be straight. The fold can be anywhere - if you only need 10" fold the fabric 10" from the selvage at each end. If the measurements aren't the same you fold a little more or less on one end till they are the same. To be extra sure you could measure the center as well. I said SHOULD because that is assuming your fabric is "on grain" when you start out.